Functions, relations and graphs

Relations versus functions, the vertical line test, and how to find domain and range from a graph, using interval notation.

Relations and functions

A is any set of ordered pairs $(x, y)$. Every graph you can draw is a relation. A function is a special relation in which each $x$-value is paired with only one $y$-value.

The circle $x^2 + y^2 = 9$ is a relation but not a function: the input $x = 0$ gives two outputs, $y = 3$ and $y = -3$.

Vertical line test

Orange: the parabola y = x² − 2. Blue: the circle x² + y² = 9. Slide the pink vertical line across both graphs.

Try this

Can you find a vertical line that cuts the parabola twice? (You can't, so it is a function.)
Where does the line cut the circle twice? What does that tell you?

Domain and range

The domain is the set of $x$-values the graph uses. Read it by looking left to right. The is the set of $y$-values the graph reaches. Read it by looking bottom to top.

Inequality notation

$x \ge 2$
$-1 < y \le 4$
all real $x$

Interval notation

$[2, \infty)$
$(-1, 4]$
$(-\infty, \infty)$ or $\mathbb{R}$

Interval brackets

$$[\;]\text{ includes the endpoint}\qquad (\;)\text{ excludes it}\qquad \infty \text{ always gets } (\;)$$

Find the range from the vertex

Try this

The range of a parabola starts at the vertex. Move the sliders, then say the range out loud before you check the vertex point.

Example 1●

Find the domain and range of $y = \sqrt{x - 1} + 2$.

Domain: $x - 1 \ge 0$, so $x \ge 1$, or $[1, \infty)$.

A square root needs a non-negative inside.

$\sqrt{x-1} \ge 0$, so $y = \sqrt{x-1} + 2 \ge 2$.

√(x − 1) is never negative, and it takes every value from 0 upwards.

Range: $y \ge 2$, or $[2, \infty)$.

Example 2●●

Find the domain and range of the semicircle $y = -\sqrt{16 - x^2}$, and say whether it is a function.

$16 - x^2 \ge 0$, so $x^2 \le 16$ and $-4 \le x \le 4$. Domain: $[-4, 4]$.

Inside the root must be at least zero.

Range: $-4 \le y \le 0$, or $[-4, 0]$.

The root gives 0 to 4. The minus sign flips this to −4 to 0, so it is the bottom half of a circle of radius 4.

It passes the vertical line test, so it is a function.

Each x gives only one y because we have chosen one half of the circle.

Flashcards

Relation

Any set of ordered pairs $(x, y)$.

Vertical line test

A graph is a function if every vertical line meets it at most once.

Domain

All the $x$-values used (read left to right).

Range

All the $y$-values reached (read bottom to top).

$[a, b)$ means?

$a \le x < b$: includes $a$, excludes $b$.

Practice

Which of these is a relation but not a function?
The range of $y = (x - 3)^2 + 1$ is:
Write $-2 \le x < 5$ in interval notation.
Is $x = 3$ (a vertical line) a function?
The range of $y = x^2 + 3$ is:
The domain of $y = \sqrt{x + 2}$ is:
The range of $y = \sqrt{25 - x^2}$ is:
$y = \dfrac{1}{x}$ takes every real value except $y = \ ?$